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Hi
Just finished updating the comparison between Maple 17.02 and Mathematica 9.01 in solving the 1390 Ordinary Differential Equations (ODEs) of Kamke's book:

  • Mathematica solved 80% in 7 hours and 8 minutes
     
  • Maple solved 97.5% in 43 minutes

While trying to solve the whole set, Mathematica hanged with 90 of these ODEs while Maple hanged with 6 ODEs. A pdf with a summarizing table and all the details is linked below

It is also relevant here that Maple's dsolve has close to half of its code implementing more modern methods, not found in Kamke, illustrated in the Maple 'what's new in DEs' help pages of the last 10 releases; for these other kinds of equations the difference is more impressive. I'll see to prepare another post about that.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

Comparison_Kamke.pdf

The question was "Let X be the random variable uniformly distributed in the disk centered at the origin O(0,0) with radius 1 and let Y be the random variable uniformly distributed in the square having its vertices A(6,-1), B(9,-2), C(8,-5), and E(5,-4). What is the PDF of the distance between X and Y? Is it possible to find that with Maple?"
Having a long think about the topic, I draw the conclusion that the exact closed form of the PDF/CDF, even the one can be found, would be useless because of its complexity.
Thus, an approximate formula for the CDF/PDF under consideration is a proper way. That formula can be derived in such a way. First,rotating the picture, we may consider the square having its sides horizontal or vertical: K((1/5)*sqrt(1410)-(1/2)*sqrt(10),1/sqrt(10)), L((1/5)*sqrt(1410)+(1/2)*sqrt(10),1/sqrt(10), M((1/5)*sqrt(1410)+(1/2)*sqrt(10),1/sqrt(10)-sqrt(10)), Q((1/5)*sqrt(1410)-(1/2)*sqrt(10),1/sqrt(10)-sqrt(10)). The geometry behind that is omitted.
We randomly choose a point P1 belonging to the square [-1,1] x [-1.1]. If the one belongs to the disk {(x,y):x^2+y^2 <=1}, then we randomly choose a point P2 from the square [K,L] x [Q,L]. Next, we calculate the distance between P1 and P2 (The  LinearAlgebra[Norm] command http://www.maplesoft.com/support/help/Maple/view.aspx?path=LinearAlgebra/Norm is used to this end.) and add it to the set S.This is repeated 2*10^4 times.

Converting S to an Array A, we constuct the empirical distribution X by A and find its mean mu and standard deviation sigma.


7.67568900820260

1.029831470

Let us compare the obtained empirical distribution and the normal distribution with the parameters mu and sigma.

The plot suggests a good fit between these. However, it is only semblance. Applying the Kolmogorov-Smirnov test (for example, see  http://www.mapleprimes.com/posts/119903-The-KolmogorovSmirnov-Test
), we  calculate

and

3.32619143372726

while the critical value equals 1.358098639 at the level 0.05. Thus, the hypothesis about the concordance should be rejected.

Also we draw the approximation to the PDF:


CDF.mw

This blog post is a response to a post on MaplePrimes.  MaplePrimes user wkehowski asked how the Task Model could be used to filter combinations.  The basic problem is formalated like this:  We want a function, I'll call it FilterComb, that accepts positive integers...

Himmelblau.mwOn the basis of Dragнilev method…Is there anyone interested in the algorithm to reduce the distance between the points of the given constraints? The algorithm is adapted for use in R ^ n. This is an example of its work on the surface:                      f = - (x1 ^ 2 x2-.3) ^ 2 - (x1 x2 ^ 2-.7) ^ 2 - 5;          ...

I have located a claimed webpage http://www.advanpix.com/2013/10/03/advanpix-vs-maple-sparse-solvers-comparison/ that shows Advanpix doing sparse matricies much faster than Maple.

The slowness is usually the result of poor coding or someone not well versed in Maple software. 

Anyone care to comment on the times?  I am sure the presented code there can be improved.

sin(10^-k)...

October 14 2013 Marvin Ray Burns 485 Maple

There seems to be patterns for sin(10^-k) for rational k;

Here we have the "floats."

n sin(10^(-n-1/2))

1 0.03161750640

2 0.003162272390

3 0.0003162277607

4 0.00003162277660

5 0.000003162277660

6 0.0000003162277660

7 0.00000003162277660

 

More later on using the mantissa. You're welcome to join me.

I define a partial repeating decimal as shown in the following example: if you have the decimal expansion 0.1728394877777777777777777777771939374652819101093837... 7 is called a partial repeating decimal.  

Back in 2000 I noticed a pattern in the decimal expansions of sin(10^-n) for growing n. Here is table of some integer n:

n                sin(10^-n)

1 9.98334166*10^-2

Back in 2000 I published A034948A036663, and A036664 in Sloane's Integer Sequences, now OEIS.

But today I decided to find the exact values of some such quotients.

1/9801=0.repeating(000

100010203040506070809101112131415161718192021222324252627282930313233

Greetings to all.

I have been using the numtheory package for quite some time now and it has helped me advance on a number of problems. Recently an issue came to my attention that I have known about for a long time but somehow never realized that it can be fixed. This is the fact that the numtheory package does not know about Dirichlet series, finite and infinite. Here are two links:

We’ve recently added a new set of questions to the Maple T.A. Cloud for English language proficiency tests. These questions demonstrate how Maple T.A. can be used to generate text-based questions that take advantage of the randomization feature. These questions were created by Metha Kamminga, an Independent Learning Professional in the Netherlands. Metha is a strong proponent of Maple T.A. in Europe, and transformed the testing and assessment system in Delft University before her retirement.

TU Delft University aims to transform learning through the use of technology. Its ambition is to eventually offer fully digitalized degree programs and it believes that digital testing and assessment can play an important role within this process. They are using Maplesoft’s online testing and assessment suite, Maple T.A., to move their courses to a digital assessment environment. To read the full user story, click here.

Visit the Maple T.A. Cloud to access the questions mentioned above and to browse the full collection of questions.

Fourteen Clickable Calculus examples have been added to the Teaching Concepts with Maple area of the Maplesoft web site. Four are sequence and series explorations taken from algebra/precalculus, four are applications of differentiation, four are applications of integration, and two are problems from the lines-and-planes section of multivariate calculus. By my count, this means some 111 Clickable Calculus examples have now been posted to the section.

A kryptarithm - this is an example of arithmetic, in which all or some of the digits are replaced by letters. The rule must be satisfied: the different letters represent different digits, the identical letters represent the identical digits. See the link  http://en.wikipedia.org/wiki/Verbal_arithmetic.

The following procedure, called  Ksolve , solves kryptarithms in which are used four operations ...

Hi

It's been 1 and 1/2 months since updates for the Maple Physics package have been distributed in the Maplesoft webpage "Maple Physics: Research & Development".  The number of Mapleprimes Physics posts that got rapidly addressed in this way is already large, some of them are listed here.

The experience has been great. Suggestions are implemented and problems are fixed in a couple of days since they were posted here, and the changes are made available to everybody right away. This is moving the focus of developments into the topics people are actually working on, with feedback and related downloadable updates happening every week.

The recent Physics updates are mostly related to quantum mechanics, an advanced topic, but part of the resulting functionality is interesting for algebraic computations in general. To mention but one: the "automatic combination of products of powers of same base" is now optional.

Recalling, by default, in Maple, if you enter xn xm, in order to receive x(n+m). you need to use the combine command (the same happens with products of exponentials). The idea behind the Maple approach is to give you more control over the steps. On the other hand, depending on your problem, the automatic combination of powers of the same base is a desired automatic simplification - this is for instance the Mathematica approach.

In today's update of Physics, a new Setup option, 'combinepowersofsamebase', is implemented, so that this automatic simplification is now optional. If set to true (> Setup(combine = true))you enter xn xm or exp(A) exp(B) and you respectively receive x(n+m) and exp(A+B). Being able to turn this automatic simplification ON and OFF comes in handy in varied situations.

Those more familiar with noncommutative objects (e.g. Matrices), also know that the combination of exp(A) exp(B) is not valid when the exponents A and B do not commute, unless A and B commute with their commutator AB - BA, in which case the combination can be done using Glauber's formula (also related to Hausdorff's formula). All of these cases have been implemented too.

In summary, in the latest update of Physics the combination and expansion of powers and exponentials using combine and expand now takes into account the noncommutative character of the exponents and the value of their commutator, and there is the option of having this combination happening automatically, for both noncommutative and commutative algebraic objects in general.

Other relevant changes more related to Physics are described in the PhysicsUpdates.mw distributed inside the zip that contains the updated Physics.mla linked in the "Maple Physics: Research & Development" webpage.

Mellin Transform....

September 27 2013 mriedel 255 Maple

Greetings to all. I will keep this brief and to the point. I would like to point out a certain deficit in the integral transform package. I have recently been calculating some Mellin transforms at this link and the base functions are of the following type.

g := (p, q) -> 1/(x+p)/(x+(p*q-1)/q);

Now to see the deficit here are some Mellin transforms that...

I just wanted to remind everyone that this quarter's Möbius App Challenge closes Sept. 30.  This quarter's prize is an iPad Prize Pack, which looks very cool but sadly, I'm not allowed to enter.

To enter the contest, all you need to do is:

1) Create an interactive App in Maple

2) While in Maple, log-in to the MapleCloud through the MapleCloud palette.

3) Click on the Send Document to the Cloud button

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